Interest Rates and Forward Rate Agreements
Read Hull Chapter 4
Outline
 Interest rate types
 Measuring interest rates
 Continuous compounding
 Zero rates / zero curve
 Bootstrap
 Forward rate
 Forward rate agreement (FRA)
 Term structure of interest rates
Interest Rate Types
 Common interest rates
 Treasury rates
 Rates on government instruments in its own currency
 London Interbank Offered Rate (LIBOR) rates
 LIBOR is the rate of interest at which a bank is prepared to deposit money with another bank.
 The second bank typically possesses an AA rating
 The British Bankers Association compiles LIBOR once a day on all major currencies for maturities up to 12 months
 The London Interbank Bid Rate (LIBID)
 LIBID is the rate which a AA bank is prepared to pay on deposits from another bank
 Repurchase agreement (Repo) rates
 Repurchase agreement is an agreement where a financial institution that owns securities agrees to sell them today for X and buy them back in the future for a slightly higher price, Y
 Invented to circumvent regulations
 U.S. banks could not pay interest on business checking accounts
 The financial institution obtains a temporary loan and the securities become the collateral
 The interest equals Y – X
 Known as the repo rate
 The RiskFree Rate
 Derivatives practitioners traditionally use the shortterm riskfree rate, LIBOR
 Many consider the Treasury rate is to be artificially low
 Regulations may require banks to hold highlyliquid securities
 Treasuries may have tax advantages
 Government has power to tax if it experiences budget problems
 As will be explained in a later lecture
 Financial wizards use Eurodollar futures and swapsto extend the LIBOR yield curve beyond one year
 The overnight indexed swap rate is increasingly being used instead of LIBOR as the riskfree rate
 Credit markets for LIBOR froze during the 2008 Global Financial Crisis
Measuring Interest Rates
 Compounding frequency
 The compounding frequency for an interest rate is the unit of measurement
 The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers
 Impact of Compounding
 When we compound m times per year at rate R, an amount PV grows to FV=PV(1+R/m)^{m} in one year
Compounding frequency 
Value of $100 in one year at 10% 
Annual (m=1) 
110.00 
Semiannual (m=2) 
110.25 
Quarterly (m=4) 
110.38 
Monthly (m=12) 
110.47 
Weekly (m=52) 
110.51 
Daily (m=365) 
110.516 
Continuous (m→∞) 
110.517 
Continuous Compounding
 In the limit, we obtain continuously compounded interest rates as we compound more and more frequently
 $100 grows to $100e^{RT} when invested at a continuously compounded rate R for time T
 $100 received at time T discounts to $100e^{RT} at time zero when the continuously compounded discount rate is R
 Discrete
 Future value (FV)
 Present value (PV)
 FV=PV(1 + R/m)^{m t}
 m – how many payments per year
 Continuous compounding
 Operations with e
 Future value
 e^{Rt} similar to (1 + R)^{t}
 Present value
 e^{Rt} = 1 / e^{RT} similar to (1+R)^{t} = 1 / (1+R)^{t}
 Derivatives and Forward Rate Agreements (FRA) use continuous interest rates
 Conversion Formulas
 Continuously compounded ratem, R_{c}
 Same rate with compounding m times per year, R_{m}
 R_{c}=m ln(R_{m} / m+1)
 R_{m}=m (e^{Rc / m}–1)
 Examples
 10% with semiannual compounding equals 2 ln( 0.1 / 2 +1 ) = 9.758% with continuous compounding
 8% with continuous compounding equals 4 ( e^{0.08/4}–1 ) = 8.08% with quarterly compounding
 Rates used in option pricing are nearly always expressed with continuous compounding
 Easy to derive formula
 Set FV = PV (1 + R_{m}/m)^{t m} = PV e^{t Rc}
 Trick question on exams
 Problem gives interest rates that are not continuously compounded
 Must convert interest rates to continuous compounding
Zero Rates / Zero Curve
 A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T
 Zero curve – treats bonds as if they make one payment at maturity
 Must convert coupon bonds into discount bonds or zero coupon bonds
 Example of bond pricing is below
Maturity
(years) 
Zero Rate
(% cont. comp.) 
0.5 
5.0 
1.0 
5.8 
1.5 
6.4 
2.0 
6.8 
 Bond Pricing
 To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate
 A bond has cash flows at different times
 Use appropriate interest rate to calculate present value
 In our example, the theoretical price of a twoyear bond providing a 6% coupon semiannually is
PV=3e^{0.05x0.5}+3e^{0.58x1.0}+3e^{0.064x1.5}+103e^{0.068x2.0}
PV=98.39
 Bond Yield
 The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond
 Suppose that the market price of the bond in our example equals its theoretical price of 98.39
 The bond yield is solved by finding the value for y with continuous compounding
 Can solve iteratively
 Set y to any number such as y =0.1
 If the PV of cashflows equal 98.39, then you are done.
 If PV is higher than 98.39, then select a lower number such as y = 0.08
 Then repeat until PV of cash flows equals 98.39
 Excel goal seeker can solve these problems
3e^{0.5y}+3e^{1.0y}+3e^{1.5y}+103e^{2.0y}=98.39
 Par Yield
 The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value.
 In our example we solve for c, which equals c=6.87, semiannual compounding
 Par Yield
 If m is the number of coupon payments per year
 d is the present value of $1 received at maturity
 A is the present value of an annuity of $1 on each coupon date
 Example
 m = 2
 d = 0.87284
 A = 3.70027
 Refer to derivation below. Notice where the terms m, d, and A come from.
The Bootstrap Method
 Data to Determine Zero Curve
 Half the stated coupon is paid each year*
Bond Principal 
Time to Maturity (yrs) 
Coupon per year ($)* 
Bond price ($) 
100 
0.25 
0 
97.5 
100 
0.50 
0 
94.9 
100 
1.00 
0 
90.0 
100 
1.50 
8 
96.0 
100 
2.00 
12 
101.6 
 Bootstrap – method to fill in missing values
 The bond holder earns $ 2.5 on the 97.5 during 3 months.
 The 3month interest rate is 2.5/97.5 = 2.56
 Convert to annual with quarterly compounding 2.56% x 4 = 10.256%
 This is 10.127% with continuous compounding
 Similarly the 6 month and 1 year rates are 10.469% and 10.536% with continuous compounding
 To calculate the 1.5 year rate we solve
4e^{0.10469x0.5}+4e^{0.10536x1.0}+104e^{1.5R}=96
3.7960+3.600+104e^{1.5R}=96
104e^{1.5R}=88.604
104e^{1.5R})= / 104 = 88.604 / 104
ln( e^{1.5R}) = ln (0.8520)
1.5R = 0.1602
R = 0.1068
 R = 0.10681 or 10.681%
 Similarly the twoyear rate is 10.808%
 Zero Curve Calculated from the Data
Forward Rates
 The forward rate is the future zero rate implied by today's term structure of interest rates
 Term structure of interest rates is what we calculated in last slide
 Formula
 Suppose that the zero rates for time periods T_{1} and T_{2} are R_{1} and R_{2} with both rates continuously compounded.
 Note – forward rate is a contract for an interest rate for a future time period between T_{1} and T_{2}.
 The formula becomes an approximation when rates are not expressed as continuous compounding
 The forward rate for the period between times T_{1} and T_{2} is
 Calculate the forward rates for the interest rates in the table
Year (n) 
Zero rate for nyear investment (% per annum) 
Forward rate for nth year (% per annum) 
1 
3.0 
 
2 
4.0 
5.0 
3 
4.6 
5.8 
4 
5.0 
6.2 
5 
5.5 
7.5 
 For an upward sloping yield curve:
 Forward Rate > Zero Rate > Par Yield
 For a downward sloping yield curve
 Par Yield > Zero Rate > Forward Rate
Forward Rate Agreement
 A forward rate agreement (FRA) is an agreement that a certain interest rate will apply to a certain principal during a certain future time period
 Example – you are buying or selling an interest rate for a specific time period
 Since interest rates do not have a physical existence, parties settle differences in cash
 An FRA is equivalent to an agreement where interest at a predetermined (fixed) rate, R_{K} is exchanged for interest at the market rate
 An FRA can be valued by assuming that the forward LIBOR interest rate, R_{F}, will be realized
 We are using the forward rate to make a prediction
 How else could we place a value on a future interest we do not know
 Thus, the value of an FRA is the present value of the difference between one party receiving a variable interest rate R_{F} and paying the fixed rate R_{K}
 The counterparty does the opposite
 The parties settle the FRA at the beginning when the FRA starts
 Zero sum game
 Example
 A company has agreed to receive 4% on $100 million for 3 months starting in 3 years
 The forward rate for the period between 3 and 3.25 years is 3%
 Forward rate forecasts the variable rate
 FV_{FRA}=(0.04/4 – 0.03/4)x$100 million = $250,000
 The value of the contract to the company is +$250,000 at 3.25 years
 Discount today to get present value
 Suppose rate proves to be 4.5% (with quarterly compounding
 We know the real variabe rate
 FV_{FRA}= (0.04/4 – 0.045/4)x$100 million = $125,000
 The payoff is –$125,000 at the 3.25 year point
 This is equivalent to a payoff of –$123,609 at the 3year point
 $125,000 e^{0.04475x0.25} = $123,609.36
 Continuous interest rate = 0.04475
 Remember  parties settle FRA at the start of the FRA in cash
 We are at year 3.25 and discount back by 0.25
 Could also use $125,000 / (1 + 0.045 / 4)^{0.25}
Term Structure of Interest Rates
 Theories try to explain why the yield curve is usually upward sloping
 Expectations Theory:
 Longterm interest rates should reflect expected shortterm interest rates
 forward rates equal expected future zero rates
 Market Segmentation:
 Short, medium and long rates are separate markets with their own independent supply and demand
 Liquidity Preference Theory:
 Investors prefer to invest with liquid securities but will invest in longer maturities with a premium
 Forward rates higher than expected future zero rates
 Suppose that the outlook for rates is flat and you have been offered the following choices
 Which rate would you choose as a depositor? Which maturity for your mortgage?
Maturity 
Deposit rate 
Mortgage rate 
1 year 
3% 
6% 
5 year 
3% 
6% 
 Liquidity Preference Theory continued
 To match the maturities of borrowers and lenders a bank has to increase long rates above expected future short rates
 In our example the bank might offer
Maturity 
Deposit rate 
Mortgage rate 
1 year 
3% 
6% 
5 year 
4% 
7% 
